New Optimization Methods Aim to Simplify Complex Tensor Networks

Researchers have introduced a set of numerical optimization techniques designed to make tensor networks easier to work with by reducing their internal complexity. Tensor networks are compact, scalable representations of high-dimensional data, and they underpin efficient computation in fields ranging from quantum physics and numerical partial differential equations to machine learning. The new work focuses on a task known as tensor disentangling: identifying transformations that reduce the bond dimensions of a network by exploiting its gauge freedom.

The team, led by Julia Wei, Alec Dektor, Chungen Shen, Zaiwen Wen and Chao Yang, formulated disentangling as an optimization problem over orthogonal matrices acting on a single tensor’s indices. The goal of these transformations is to minimize the rank of the tensor’s matricized form, which in turn lowers the bond dimensions needed to represent the network accurately.

To solve this problem, the researchers present Riemannian optimization methods, which operate on the manifold of orthogonal matrices. They also introduce a joint optimization framework that alternates between two steps: optimizing the orthogonal transformation while holding a fixed low-rank approximation, and then optimizing the low-rank approximation for a fixed orthogonal transformation. According to the paper, this alternating approach offers a competitive alternative when the target rank of the approximation is already known.

In many practical settings, however, the optimal rank is not known in advance. To address this, the authors propose a binary search strategy integrated with the disentangling procedure, allowing the method to seek out the often unknown optimal rank rather than requiring it as an input. This combination of search and optimization is intended to make the technique usable in scenarios where the appropriate level of compression must be discovered automatically.

The methods were evaluated in numerical experiments on random tensors and on tensors drawn from an approximate isometric tensor network state. These experiments compared the performance of the different optimization methods against one another and explored whether combining them in a hybrid approach could yield further benefits. The study appears as an open access research article in Springer Nature’s Computational Science and Engineering collection, published on 16 September 2026, and is currently shared as a peer-reviewed, accepted version of record subject to further edits.

Subject of Research: Numerical optimization methods for tensor disentanglement in tensor networks

Article Title: Numerical optimization for tensor disentanglement

Article References: Wei, J., Dektor, A., Shen, C., Wen, Z., & Yang, C. (2026). Numerical optimization for tensor disentanglement. Computational Science and Engineering. https://doi.org/10.1007/s44207-026-00018-7

Image Credits: AI Generated

DOI: 10.1007/s44207-026-00018-7

Keywords: tensor networks, tensor disentanglement, bond dimension, Riemannian optimization, orthogonal matrices, low-rank approximation, binary search, gauge freedom, numerical PDEs, machine learning, quantum physics, Numerical